where again the \(h\) - (respectively \(e\) -) determinants are \(n\times n\)  (respectively \(m\times m\) ) if \(\ell (\lambda ) \le n\)  (respectively \(\lambda _1 \le m\) ).
We will prove the first identity, and make remarks about how to prove the second. The remaining two are completely analogous and we omit the details. The idea, following Gessel [1]} and Tracy–Widom [2]}, is to use the Cauchy–Binet identity. That is, let \((A_{i,j})_{1 \le i \le n; 1 \le j < \infty }\)  be the \(n \times \infty \)  matrix \(A_{i,j} = h_{j-i} (\rho ^-)\) , and \((B_{i,j})_{1 \le i < \infty ; 1 \le j \le n}\)  be the \(\infty \times n\)  matrix \(B_{i,j} = h_{i-j}(\rho ^+) + h_{i+j-2n}(\rho ^+)\) . Then
\(\det AB = \sum _{L} \det A_L \cdot \det B_L\) 
